Equivalence of definitions of Sylow direct factor: This states that a conjugacy-closed normal Sylow subgroup is a direct factor. Note that the statement is again a weakening of the fact that any conjugacy-closed Sylow subgroup is a retract, and it again has a more direct proof.

Facts used

Proof

(2) implies (3)

(Note: This implication uses nothing about the special nature of normalizers of -subgroups, and works for all subgroups).

Given: A finite group , a prime , a -Sylow subgroup of . A normal subgroup of such that and is trivial.

To prove: If is a non-identity -subgroup of , then has a normal -complement.

Proof: Let . By the second isomorphism theorem, we have:

.

Since divides , and equals the order of , is a power of . Thus, so is the order of the right side, .

Thus, is a normal subgroup of whose order is relatively prime to (since it is also a subgroup of ) and whose index is a power of . Thus, it is a normal -complement in .

(3) implies (4)

Given: A finite group , a prime , a -Sylow subgroup of . For any non-identity -subgroup , has a normal -complement.

To prove: If is a non-identity -subgroup of , then is a -group.

Proof: Proof: Let . Let be a normal -complement in .

Now, and are both normal subgroups of and since the order of is relatively prime to , is trivial. Thus, . Thus, by fact (1):

.

Since the left side is a power of , so are both terms of the right side. In particular, is a -group.

(2) implies (1)

This follows from fact (3).

(1) implies (2)

This follows from fact (4).

References

Textbook references

Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page Theorem 4.5, Chapter 7 (Fusion, transfer and p-factor groups), warning.png",Chapter7(Fusion,transferandp-factorgroups)" is not declared as a valid unit of measurement for this property.warning.png",Chapter7(Fusion,transferandp-factorgroups)" is not declared as a valid unit of measurement for this property.warning.png",Chapter7(Fusion,transferandp-factorgroups)" is not declared as a valid unit of measurement for this property.More info